1Place the flour, sugar, yeast, and salt in the bowl of a stand mixer and whisk to combine. Add the yolks and milk and mix, using the hook attachment, on medium-low speed until a shaggy dough forms, about 1 minute. Add the butter, increase the speed to medium high, and mix until the dough is smooth, shiny, and elastic, about 5 minutes.
2Coat a large bowl with oil. Form the dough into a ball, place in the bowl, and turn to coat in the oil. Cover with plastic wrap or a damp towel and let rise in a warm place until doubled in size, about 1 to 1 1/2 hours.
3Lightly flour a baking sheet; set aside. Punch down the dough, transfer to a lightly floured work surface, and roll until about 1/4 inch thick. Using a 2-inch round cutter, stamp out as many dough rounds as possible and place on the prepared baking sheet about 1/2 inch apart. Gather the dough scraps into a ball and roll out again, stamping rounds until you have 30 total on the baking sheet. Cover loosely with plastic wrap or a damp towel. Let rise in a warm place until puffy and about 1/2 inch thick, about 30 minutes.
4Place the vegetable or canola oil in a Dutch oven or a large, heavy-bottomed pot and set over medium heat until the temperature reaches 350°F on a candy/fat thermometer. Meanwhile, line a second baking sheet with paper towels and place a wire rack over the paper towels; set aside. Place the jam or jelly in a piping bag fitted with a 1/4-inch round tip; set aside.
5Using a flat spatula (don’t use your hands—this will deflate the donuts), carefully transfer the dough rounds, one at a time, into the oil. You should be able to fit about 6 at a time, leaving at least 1 inch of space in between and keeping the oil temperature at 350°F. Fry until the bottoms are golden brown, about 1 1/2 minutes. Carefully flip with a fork and fry until the second side is golden brown, about 1 1/2 minutes more. (If air bubbles appear in the donuts, pierce with the tip of a paring knife.) Remove with a slotted spoon to the wire rack. Repeat with the remaining dough rounds.
6When the donuts are cool enough to handle, use a paring knife to puncture the side of each to form a pocket in the center. Place the tip of the piping bag into the pocket and pipe about 1 teaspoon of jam or jelly inside. Dust with powdered sugar before serving.